Sample Problem: Torsional Stress Calculation of a Stepped Shaft
The stepped shaft shown in the figure is to rotate at 900 rpm as it transmits 7000 Nm
torque from a turbine to a generator and this is the only loading case on the shaft. The material specified in the
design is A 284 Steel (grade C) and design factor is given as 2. Determine/evaluate following cases for the shaft.
a) Maximum shear stress on the shaft
b) Principal stresses on the shaft
c) Material yield criteria for selected material and occurred stresses.
Solution:
Step 1 : Write down input parameters (including material properties) which are
defined in the sample example.
INPUT PROPERTIES SUMMARY 
Parameter 
Symbol 
Value 
Unit 
Diameter of larger shaft section 
D 
100 
mm 
Diameter of smaller shaft section 
d 
50 
mm 
Radius of smaller shaft section 
c_{2} 
25 
mm 
Torque 
T 
7000 
Nm 
Rotation speed 
w 
900 
rpm 
Design factor 
n_{d} 
2 
 
Yield Strength (A284 Steel) 
Sy 
205 
MPa 
Elastic modulus(A284 Steel) 
E 
140 
GPa 
Shear (Rigidity) modulus(A284 Steel) 
G 
80 
GPa 
Elongation at break(A204 Steel) 
ε_{brk} 
23% 
 
Step 2 : Go to "Torsion of Solid and Hollow Shafts Calculator" page to calculate maximum shear
stress on the shaft. Larger shear stresses occur on smaller diameter section of the shaft so analysis of smaller diameter section is sufficient for this example.
RESULTS 
Parameter 
Symbol 
Value 
Unit 
Maximum shear stress 
τ_{max} 
285.206 
MPa 
Angle of twist 
_{θ} 
4.085 
Degree 
Power requirement 
P 
659.734 
kW 
Polar moment of inertia 
J 
613592.312 
mm^4 
Step 3 : There is a shoulder fillet in the shaft design and this geometry will
raise stress . Stress concentration factor and maximum shear stress
for shoulder fillet will be calculated for torsional loading . Go to "Shoulder fillet
in stepped circular shaft" page for calculations.
LOADING TYPE  TORSION 

Parameter 
Symbol 
Value 
Unit 
Stress concentration factor

K_{t} 
1.25 
 
Nominal shear stress at shaft 
τ_{nom} 
285.21 
MPa 
Maximum shear stress due to torsion 
τ_{max} 
357.03 
Maximum shear stress of 357 MPa occurred at outer radius of shoulder
fillet. This is the answer of clause a) of the sample example.
Step 4 : To calculate principal stresses occurred on the shaft, go to the
"Principal/Maximum Shear Stress Calculator For Plane Stress" page. Note that the torsional
loading of shaft results plane stress state on the surface of shaft so this calculator
can be used.
RESULTS 
Parameter 
Symbol 
Value 
Unit 
Maximum principal stress 
σ_{max} 
357 
MPa 
Minimum principal stress 
σ_{min} 
357 
Maximum shear stress* 
τ_{max} 
357 
Average principal stress 
σ_{avg} 
0 
Von Misses stress 
σ_{mises} 
618.3 
Angle of principal stresses ** 
θ_{p} 
45 
deg 
Angle of maximum shear stress ** 
θ_{s} 
0 
Principal stresses are calculated as 357 MPa and 357 MPa. This is the
answer of clause b) of the sample example.
Step 5 : Selected material (A284 Steel) is ductile since elongation at break is
greater than 5%. For the evaluation of yield criteria for a ductile material
with plane stress state , we can use "Yield Criteria For Ductile Materials Under
Plane Stress(Static Loading)" page.
RESULTS 
Parameter 
Condition to be met for safe design 
Status 
MSS theory 
(σmaxσmin) < Sy/n 
714<102.5 
Nok 
DE theory 
(σmax^2σmax*σmin+σmin^2)^0.5<
Sy/n 
618.3 < 102.5 
Nok 
Summary
According to results, the design is not
safe for the given parameters and conditions. Shaft diameter or material
shall be changed to satisfy required design criteria. Steps listed above shall
be repeated to find dimensions or materials that satisfy
required conditions.
Note: In this example, the loading case
is static and shaft material is ductile. According to Shigley's Mechanical Engineering Design Chapter 3 ,
for ductile materials in static loading, the stressconcentration factor is not usually applied to predict the
critical stress, because plastic strain in the region of the stress is localized and
has a strengthening effect.
According to Peterson's Stress Concentration Factors
Chapter 1, the notch sensitivity q usually lies in the range of 0 to 0.1 for
ductile materials. If a statically loaded member is also subjected to shock
loading or subjected to high and low temperature, or if the part contains sharp
discontinuities, a ductile material may behave like a brittle material. These
are special cases and if there is a doubt, K_{t} (q=1) shall be applied.
In this example, since there is no information
about temperature and shock loading condition of the shaft,
the notch sensitivity factor q is taken as 1 and K_{t} is applied .
The problem is fully solved with calculators which are summarized as
follows.